Gradient Plasticity Model and its Implementation into MARMOT · 2014-06-06 · Gradient Plasticity...

Gradient Plasticity Model and its Implementation into MARMOT

PNNL-22702

Prepared for

U.S. Department of Energy

Nuclear Energy Enabling Technology Program

Erin Barker, Dongsheng Li, Hussein Zbib, Xin Sun

Pacific Northwest National Laboratory, Richland, WA 99352

August 2013

NEAMS Milestone Report M3MS-13PN0602051

DISCLAIMER This information was prepared as an account of work sponsored by an agency of the U.S. Government. Neither the U.S. Government nor any agency thereof, nor any of their employees, makes any warranty, expressed or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness, of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. References herein to any specific commercial product, process, or service by trade name, trade mark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the U.S. Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the U.S. Government or any agency thereof.

Gradient Plasticity and its implementation into MARMOT Aug 2013 iii

SUMMARY

The influence of strain gradient on deformation behavior of nuclear structural materials, such as body-‐centered cubic (bcc) iron alloys has been investigated. We have developed and implemented a dislocation based strain gradient crystal plasticity material model. A mesoscale crystal plasticity model for inelastic deformation of metallic material, bcc steel, has been developed and implemented numerically. Continuum Dislocation Dynamics (CDD) with a novel constitutive law based on dislocation density evolution mechanisms was developed to investigate the deformation behaviors of single crystals, as well as polycrystalline materials by coupling CDD and crystal plasticity (CP). The dislocation density evolution law in this model is mechanism-‐based, with parameters measured from experiments or simulated with lower-‐length scale models, not an empirical law with parameters back-‐fitted from the flow curves.

In our current framework, geometrically necessary dislocations are introduced to take into consideration of strain gradient for the long range interactions. Two approaches have been proposed to incorporate the influence of strain gradient into the framework: the first one with analytical solution in a homogenization method, i.e., viscoplastic self-‐consistent (VPSC) model, the other one with user material subroutine in finite element method (FEM).

The mesoscale plasticity model is formulated to treat both long-‐range and short-‐range processes and interactions. Models for the evolution of mobile and immobile dislocations, as well as interstitial loops, and interaction hardening laws, are formulated based on quantifiable mechanisms from lower length scales, such as dislocation multiplication, annihilation, junction formation/breakage, and cross-‐slip in CDD. Long-‐range interactions, resulting from dislocation structures, will be treated within a dislocation-‐based strain gradient theory, compatible with dislocation theory and driven by densities represented as continuum fields.

Crystal plasticity has been implemented into MARMOT to increase the capability of the numerical solution framework for mechanical deformation behavior of nuclear structural materials. Combined with the lower length scale simulation capability in MOOSE, development in this capability will build up a multi-‐scale and multi-‐physics modeling architecture.

Gradient Plasticity and its Implementation into MARMOT iv Aug 2013

CONTENTS

1. Introduction .................................................................................................................................. 1 2. Strain Gradient Continuum Dislocation Dynamics ..................................................................... 5 3. Strain Gradient Field Calcualted from Polycrystalline Viscoplasticity Model ......................... 13 4. Results and Discussion .............................................................................................................. 15

4.1 Low Length Scale Models Providing Dislocation Density Evolution Law ..................... 15 4.2 CDD in Predicting Mechanical Properties of Fe Single Crystal ...................................... 20 4.3 Prediction of Deformation Behavior and Texture Evolution using CDD-CP .................. 25

5. Implementation of Crystal Plasticity in MARMOT .................................................................. 30 5.1 MARMOT Implementation .............................................................................................. 32 5.2 Plan for Strain Gradient Addition to MARMOT Implementation ................................... 34

5.2.1 Theoretical Overview of Higher Order Strain Gradient Formulation ................... 38 Acknowledgements ........................................................................................................................... 41 References ......................................................................................................................................... 42

Gradient Plasticity and its implementation into MARMOT Aug 2013 v

LIST OF FIGURES Figure 1. Scheme of multi-scale approach in mechanical flow prediction ......................................... 2 Figure 2. Work flow of multi-scale modeling predicting the plastic flow with long range

interaction taken into consideration by strain gradient plasticity ................................................ 2 Figure 3. Dislocation mobility as a function of the temperature in Fe-Ni system, with

concentration of nickel or chromium was varied from 5% to 20% .......................................... 16 Figure 4. a) Dislocation-defect structure after plastic deformation. b) Stress-strain curves

simulated by DDD for iron single crystal at different defect densities (a:0, b:2×1022, c:5×1022, d: 7×1022, e: 1×1023 , f: 2×1023.) ................................................................................ 17

Figure 5. Discrete dislocations dynamics results: Evolution of defect density (a) and dislocation density (b) for iron alloy with 5%Ni concentrations at different initial irradiation defect density: (a:0, b:2×1022, c:5×1022, d: 7×1022, e: 1×1023 , f: 2×1023.) , (c) evolution of total dislocation density, cross-slip segments and dislocations junctions. ........................................ 19

Figure 6. Comparison of experimental stress strain curves and simulated results from SCCE-T, SCCE-D and CDD models for iron single crystal with uniaxial tensile loading direction along (a) [100] (b) [011] and (c) [-348] directions. The constitutive law parameters for SCCE-T and SCCE-D are back-fit to the [100] results. ............................................................ 22

Figure 7. Comparison of experimental stress strain curves and simulated results from SCCE-T, SCCE-D and CDD models for iron single crystal with uniaxial tensile loading direction along (a) [100] (b) [011] and (c) [-348] directions. The constitutive law parameters for SCCE-T and SCCE-D are back fit to the [-348] results. ........................................................... 23

Figure 8. (a) 100, (b) 110, and (c) 111 pole figures of simulated bcc iron sample with random texture. ....................................................................................................................................... 25

Figure 9. Predicted stress strain curve of the random texture iron using CDD-CP model. .............. 26 Figure 10. Predicted (100), (110) and (111) pole figures of random bcc iron alloys under uniaxial

tension at strain of (a) 2% and (b) 10%. .................................................................................... 26 Figure 11. Scheme of data flow in closely coupled CDD and CP (a) without (b) with long

distance interaction considered. ................................................................................................ 27 Figure 12. Strain field of random textured polycrystalline agglomerate after uniaxial tensioned to

a strain of (a) 0.1, (b) 0.2 (c) 0.3 (d) 0.5 (e) 0.8 and (f) 1.0. ..................................................... 28 Figure 13. Strain gradient field of random textured polycrystalline agglomerate after uniaxial

tensioned to a strain of (a) 0.1, (b) 0.2 (c) 0.3 (d) 0.5 (e) 0.8 and (f) 1.0 .................................. 29 Figure 14. Strain field of random textured polycrystalline agglomerate with 2000 grains after

uniaxial tensioned to a strain of (a) 0.1, (b) 0.2 (c) 0.3 (d) 0.5 (e) 0.8 and (f) 1.0 .................... 29 Figure 15. Strain gradient field of random textured polycrystalline agglomerate with 2000 grains

after uniaxial tensioned to a strain of (a) 0.1, (b)0.2 (c) 0.3 (d) 0.5 (e) 0.8 and (f) 1.0 ............. 30

Gradient Plasticity and its Implementation into MARMOT vi Aug 2013

LIST OF TABLES

Table 1 . List of parameters used in the continuum dislocation dynamics model ............................ 20 Table 2. Variance of the simulation results from experimental stress strain curve for two CDD

parameters ................................................................................................................................. 24

Gradient Plasticity and its implementation into MARMOT Aug 2013 vii

ACRONYMS

BCC body-‐centered cubic

CDD continuum dislocation dynamics

CP crystal plasticity

Cr chromium

CRSS critical resolved shear stress

DD dislocation dynamics

DDD discrete dislocation dynamics

DOE U.S. Department of Energy

FCC face-‐centered cubic

Fe iron

FS Frank sessile

GND geometrically necessary dislocation

MD molecular dynamics

MWLSR moving weighted least squares regression

Ni nickel

PNNL Pacific Northwest National Laboratory

R&D research and development

SCCE-‐D single crystal constitutive equations, based on dislocation density

SCCE-‐T single crystal constitutive equations, for texture analysis

SSD statistically stored dislocations

UMAT user defined material routine

VPSC viscoplastic self consistent

Gradient Plasticity and its Implementation into MARMOT Aug 2013 1

NUCLEAR ENERGY ADVANCED MODELING AND SIMULATION PROGRAM

Gradient Plasticity Model and its Implementation into

MARMOT

1. INTRODUCTION Plastic flow strength of nuclear structural materials depends on irradiated

microstructures and other inherent properties. From the perspective of strain, it depends on

strains and strain gradients. In the theory of dislocation dynamics, dislocations from plastic

deformation may be categorized into: statistically stored dislocations (SSD) from bulk plastic

strain and geometrically necessary dislocation (GND) from strain gradient. The strain gradient is

attributed to the heterogeneity of the microstructure and geometric shape and size. In the

early days of the gradient plasticity theory development, most of the studies focused on

geometrically heterogeneous systems, such as sharp crack tip, thin wire torsion, indentation

tests and so on.

In this study, the strain gradient will be focused on the heterogeneity from inherent

microstructure, such as texture, precipitate, voids, and other heterogeneous microstructure

contribution. The strain gradients are inversely proportional to the length scale of investigated

system when the difference of strain is constant. Consequently, the contribution from strain

gradient becomes apparent when the investigated length scale is small enough to reach the

range of micrometer. Phenomenological theories of strain gradient plasticity were proposed by

Fleck, Hutchinson and other front runners in the past decades, as summarized in several

excellent reviews (Evans and Hutchinson, 2009; Fleck and Hutchinson, 1997; Fleck et al., 1994).

In our previous studies (Li et al., in preparation; Zbib et al., 2012), we have developed a

multi-‐scale modeling framework to simulate the mechanical deformation behavior of nuclear

structural materials including bcc iron alloys. The work flow of this multi-‐scale framework starts

Gradient Plasticity and its Implementation into MARMOT 2 Aug 2013

from molecular dynamics (MD), to discrete dislocation dynamic (DDD), to continuum

dislocation dynamics (CDD), to crystal plasticity (CP) model and FEM.

Figure 1. Scheme of multi-scale approach in mechanical flow prediction

Scale bridging in this framework has two types, tightly-‐coupled and loosely-‐coupled. The

information pass from MD to DDD is loosely-‐coupled: dislocation mobility as a function of

temperature and alloy composition is calculated from MD and passed in a suitable form to DDD

model. The results gathered from the DDD simulations then lead to the evaluation of the

dislocation density and critical resolved shear stress in Fe-‐Ni-‐Cr alloys as well as irradiation

damage. The dislocation density evolution law is passed into CDD for mechanical behavior

prediction. The bridging between CDD and polycrystalline viscoplasticity model is a tightly-‐

coupled model. Finally, the results from CP are used as a material subroutine in FEM model.

To incorporate the influence of the strain gradient due to the sample geometry, length scale

effect and heterogeneous microstructure, gradient plasticity has been utilized and added to the

current framework. The updated workflow is illustrated in Figure 2 below:

Figure 2. Work flow of multi-scale modeling predicting the plastic flow with long range interaction taken into consideration by strain gradient plasticity


In the early stage of strain gradient plasticity model, most models are

phenomenological, treating the stress contributing the strain gradient as work conjugate higher

order stresses in the form of couple stresses and double stresses, dealing with the strain

gradient tensor in the form of deformation curvatures (Fleck and Hutchinson, 1993; Fleck and

Willis, 2009a, b). The experimental evidences for the scale effect due to the strain gradient have

been observed in the plastic flow of metals and ceramics. For example, the indentation

hardness increases by a factor of 2 as the width of the indent is decreased from about 10 um to

1um (Stelmashenko et al., 1993). This is one example of Hall-‐Petch effect, the yield strength

increases with the decrease of grain size. To capture the scale effect, or the grain boundary

effect, we incorporated gradient plasticity.

The gradient plasticity theory used in our model is different from the fore-‐mentioned

approach in two aspects: one is on the contribution of the strain gradient. In our model, two

sources of strain gradients are taken into considerations: one is from sample geometry

calculated by FEM model, the other one is from texture, calculated by polycrystalline

viscoplasticity model. The other difference is our model is mechanism-‐based, not a

phenomenological one. GND from strain gradient is incorporated in the model at the stage of

dislocation dynamics.

Before the introduction of gradient plasticity, the elastic-‐plastic formulations developed

since the 1960s are strain rate independent (Asaro and Rice, 1977; Hill, 1966; Mandel, 1965;

Simo and Taylor, 1985) with yield surface defined by Hill’s criterion. Later, a viscoplastic

approach was proposed to achieve unique solutions to avoid the numerical multiple solutions

from using strain rate sensitivity formulations (Asaro and Needleman, 1985; Hutchinson, 1976;

Peirce et al., 1982). The essence of viscoplasticity theory is the hardening law, usually in a

power law format, to relate the shear stress in each slip system to the shear strain rate. The

advantage of the power law formulation is due to the derivation of viscoplastic potential with

the capability to capture saturation effect at a high strain rate. Different versions of isotropic

and anisotropic viscoplastic constitutive theories have been developed. Examples include

MATMOD equations proposed by Miller (Henshall, 1996; Miller, 1976) to describe

viscoplasticity function as a combination of a hyperbolic sine function and a power function.


Limited to small strain, back stress is used for kinematic hardening and a drag stress for

isotropic hardening. Another example is Robinson’s formulation, which is more complicated,

using a back-‐stress, a drag stress, a yield stress, and a power function for the viscoplastic flow

(Arnold and Saleeb, 1994). There are many other versions taking into consideration different

factors. With the development of the theory of slip systems, hardening is decomposed into two

parts: latent hardening and self hardening, both in the exponential function form to describe

interaction between different slip systems (Asaro, 1983; Asaro and Needleman, 1985; Peirce et

al., 1982). Nevertheless, most theories of resistance of shear stress evolution are empirical with

the parameters determined by fitting the model to the experimental stress strain curves.

With the advent of dislocation dynamics and multi-‐scale modeling, alternative

constitutive laws were proposed to introduce the knowledge generated from dislocation theory

to the continuum plasticity framework. For example, Zbib and de la Rubia (Zbib and Diaz de la

Rubia, 2002), Devincre (Devincre et al., 2008), Groh et al. (Groh et al., 2009), and Alankar et al.

(Alankar et al., 2012b) proposed multi-‐scale approaches to establish a dislocation-‐based

continuum model to incorporate discrete and intermittent aspects of plastic flows. In these

approaches, strain hardening can be predicted through the modeling of mean free paths of

dislocations. Other statistical aspects from dislocation dynamics simulation, such as dislocation

densities, are used as internal state variables to capture deformation behavior of single crystals

(Arsenlis and Parks, 2002; Ortiz et al., 2000). Another front noteworthy is Sandfeld’s work

(Hochrainer et al., 2007; Sandfeld et al., 2011) to bridge statistical continuum mechanics with

dislocation dynamics by dislocation density tensor. Using a statistical description of dislocation

interactions in terms of a Taylor-‐type yield stress and a back stress, which describes short-‐range

repulsion of dislocations of the same sign, key features from discrete dislocation dynamics

simulations were captured. The core of the interaction between the discrete dislocation

dynamics and crystal plasticity is the evolution law of dislocation density.

Discrete dislocation dynamics (DDD) is a powerful tool that has been advanced

significantly in the past decade (Canova et al., 1993; Ghoniem and Sun, 1999; Zbib et al., 1996,

1998). It has been used to explain the effect of irradiation on mechanical properties through

large-‐scale simulations of interaction of numerous numbers of dislocations with defect clusters.


The work of Zbib and co-‐worker has shown that dislocations interactions with the elastic fields

of nano-‐sized defect clusters in Cu and Pd lead to hardening followed by localized deformation

and channel formation resulting from defect cluster annihilation by dislocations (Diaz de la

Rubia et al., 2000; Ghoniem et al., 2000; Hiratani et al., 2002a; Hiratani et al., 2002b; Khraishi et

al., 2002a; Li et al., 2010; Zbib et al., 2000). In the present work we investigate the effect of

irradiation hardening in Fe-‐Ni-‐Cr systems. The goal is to predict the stress-‐strain curve and the

critical resolved shear stress as a function of defect density, which is then used in the crystal

plasticity model. However, in order for this method to work, a detailed knowledge of the

dislocation mobility in an analytical form inside the materials is required. To effectively predict

the mechanical behavior of the materials under irradiation at the continuum scale, critical

information should be determined and progressively passed from one scale to another.

2. STRAIN GRADIENT CONTINUUM DISLOCATION DYNAMICS

The strain gradient CDD framework uses the strain gradient calculated from VPSC and FEM

model to feed back for the GND and recalculate the flow stress. The principle of crystal

plasticity was improved by introducing more physics-‐based mechanisms to substitute

empirically derived constitutive and hardening laws. Based on the traditional kinematics of

crystal plasticity, CDD also has three fundamental assumptions. First, the total deformation

gradient F (Peeters et al., 2000; Schoenfeld et al., 1995) is assumed to be the product of two

terms, elastic part Fe and plastic part Fp:

peFFF = , (1)

where, Fe is due to the elastic distortion of the lattice, and the plastic part Fp is due to the slip

by the dislocation motion in the unrotated intermediate configuration. It evolution rate, pF , is

expressed by the kinematic relation:

ppp FLF = (2)


In the second assumption, the plastic velocity gradient Lp is expressed as the sum of a number

of crystallographic slip rates, αγ :

Lp = γαnα ⊗mα

α=1

N

∑ (3)

where nα and mα are the vectors representing slip direction and slip plane normal of slip system

α, and N is the number of slip systems.

In the third assumption here, the lattice deformation is equal to the elastic material

deformation. The plastic velocity gradient consists of a symmetric part, strain rate DP, and an

antisymmetric part, total spin WP:

( )∑=

+==N

ijjipij

pij mnmnLsymmD

121)(

α

αααααγ (4)

( )∑=

−==N

ijjipij

pij mnmnLsymmW

121)(

α

αααααγ (5)

Here, the plastic strain rate DP determines the deformation behavior by updating the

stress strain curve; while the total plastic spin WP is used to update the orientation of crystals,

texture, in polycrystalline materials.

Normally, the shear strain rate of each slip system α, αγ , is determined by a strain-‐rate-‐

dependent power law function of the form:

( )αα

αα τ

ττ

γγ signm1

00 =

(6)

where 0γ is reference strain rate, ατ 0 is the slip resistance of the slip system α, ατ is the

resolved shear stress along slip system α, and m is the rate sensitivity exponent. The

parameters ατ 0 and m are obtained from experimental results using curve fitting.

In SCCE-‐T, the hardening law defines the evolution of critical resolved shear stress

(CRSS) of slip system and is defined as follows (Asaro, 1983):


∑=β

ααβ

α γτ h0 (7)

where, αβh is the hardening coefficient matrix with self-‐hardening and latent hardening

components.

In contrast, in the CDD model discussed below, another form of power law is used by

utilizing the results from DDD. Our constitutive law for the shear strain rate is based on the

Orowan relation (Orowan, 1940):

ααα ργ gMbv= (8)

where αρM is the mobile dislocation density of slip system α, b is the Burgers vector, and α

gv is

the average dislocation glide velocity in slip system α.

In previously developed models, the dislocation glide velocity αgv , is expressed in a

power law of resolved shear stress ατ similar with shear strain rate of slip system α, ατ 0 :

( )αα

αα τ

ττ signvv

m

g

1

00= (9)

Here the expression of dislocation glide velocity is improved to a general law using Kock-‐

type activation enthalpy:

v!α = v!l!exp − ∆!!!"

1− τα !τ!α

τ!α

! !sign τα

0

when τα > τ!α

when τα ≤ τ!α (10)

where lg is the distance between the barriers. For a kink height, lg is close to Burgers vector. VD

is the Debye frequency, and k is the Boltzmann constant. T is temperature, ∆F! is Kock-‐type

activation enthalpy, τ!α is the Peierls lattice resistance of the slip system α, and p and q are

strain rate parameters. In this new proposed law, all parameters have physical meaning

calculated from lower-‐length scale models. The slip resistance of slip system α, ατ 0 , is a

summation of reference resistance τ0 from lattice friction stress for moving dislocations;


resistance due to dislocation-‐defect interaction τdα, mainly from irradiation; and resistance from

dislocation hardening ατ dh :

ααα ττττ dhd ++= 00 (11)

According to the Baily-‐Hirsch model (Ohashi, 1994), the resistance from dislocation

hardening is slip resistance of statistically stored dislocations on other slip systems against

moving ones on one specific slip system α. It is a function of interaction matrix of slip systems

mαΩ :

m

m

mdh b ρµατ αα ∑ Ω=

(12)

Here, b is Burgers vector, µ is shear modulus, α is a numerical factor on the order of 0.1,

and ρm is the density of statistically stored dislocations accumulated on the slip system m.

The evolution rate of statistically stored dislocation density is related to the mean free

path of moving dislocations on slip system α, Lα (Ohashi et al., 2007):

ααα γρ bLc / = (13)

where αγ is the shear strain rate of slip system α, and c is a numerical constant of order 1. Lα is

a function of dislocation density.

Assuming defects are distributed randomly for all slip systems, a modified dispersed

barrier hardening model is used to express irradiation resistance ατ d from dislocation-‐defect

interaction:

( )ndd dd ρµβτ α = (14)

where ρd is the defect density and d is the defect size. In a traditional dispersed barrier

hardening model, n=0.5. In our proposed model, n is calculated directly from discrete DD.

In the previously proposed DD-‐based crystal plasticity models, the dislocation density

evolution laws are either ignored or used a fitted curve to represent the evolution (as proposed

by Kocks (Kocks, 1976)):


ααβ

βα γρρ

ρ ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−=

∑b

a

n

kkb

1 (15)

where ka and kb are material parameters for dislocation generation and annihilation,

respectively. This formulation was utilized by Wagoner et al. (Lee et al., 2010a) and Zbib et al.

(Zbib and de la Rubia, 2002) on flow simulation of single crystals. Simulation results from this

model will be compared with the mechanism-‐based CDD.

The evolution rate of the dislocation density is based on mechanisms that can be

quantified from the discrete DD. Generally, the statistical stored dislocations can be divided

into two types, mobile and immobile:

ααα ρρρ IMS += (16)

In CDD, the evolution of mobile dislocation density is composed of six terms with

different physical contribution. The first mechanism is from the multiplication and growth of

resident dislocations and the production of new dislocations from Frank-‐Reed sources in slip

system α:

α

ααα ραρlvg

MM 11 = (17)

where α1 is the dislocation multiplication coefficient, αρM is the mobile dislocation density

distributed on slip system α, and αl is the mean free path of dislocations on slip system α.

The second mechanism captures the mutual annihilation of two mobile edge or screw

dislocations with opposite signs in slip system α:

( ) gMcM vR 222 2 αα ραρ −= (18)

where α2 is the dislocation annihilation coefficient and Rc is the capture radius for the

dislocation annihilation event.

The third mechanism describes the transition of mobile type to immobile type due to

the interaction between dislocations:


lvg

MMαα ραρ 33 −= (19)

where α3 is the immobilization parameter.

Conversely, the fourth mechanism is about the mobilization of immobile dislocations

due to the breakup of junctions, dipoles, pinning parts, etc., at a critical stress condition:

lv g

I

r

M

αα

α

α ρτ

ταρ

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛= *44 (20)

where α4 is the mobilization parameter.

The fifth mechanism considers cross-‐slip, the phenomena where screw dislocation

segments on one slip plane move to another glide plane during plastic deformation. Extensive

reviews of cross-‐slip in experiments and simulations have been given by Jackson (Jackson, 1985)

and Püschl (Püschl, 2002). Cross-‐slip has been observed in experimental work and simulated

using molecular dynamics, dislocation dynamics, and other low-‐length scale models (Alankar et

al., 2012a; Bitzek et al., 2008; Püschl, 2002). In large-‐length scale, cross-‐slip was incorporated in

a continuum model in an empirical way, either assuming a recombination distance or a

dissociation energy (Püschl, 2002). There is a disconnection in the modeling influence of cross-‐

slip on deformation behavior across different length scales. In most continuum texture models,

cross-‐slip has been ignored. In our model, evolution of mobile dislocation due to cross-‐slip is

defined as below:

lv

P gN

MM

α

β

βαβα ραρ ∑=

=1

55 (21)

where α5 is the cross-‐slip parameter, N is the number of cross-‐slip planes available for each

Burgers vector, e.g. in fcc N=2, and in bcc N=12. αβP is the cross-‐slip probability matrix; the

components of this matrix are either 0, -‐1 or +1; 0=αβP means no cross-‐slip of dislocations

from system β to system α , 1+=αβP indicates cross-‐slip of dislocation from β to α , and

1−=αβP means that the system α lost dislocations to system β by cross-‐slip. In this setup,


cross-‐slip is treated as a stochastic process and, therefore, we use a Monte-‐Carlo type analysis

similar to that used in the discrete dislocation dynamics for cross-‐slip (Rhee et al., 1999; Zbib et

al., 1998). We emphasize the introduction of this stochastic term that makes it possible to

predict the anisotropic behavior of single crystals for different loading directions as shown in

the next section. More details regarding the implementation of cross-‐slip in the continuum

dislocation theory are given in Li et al (Li et al., in preparation).

The sixth mechanism is similar to the second one, also annihilation, but between mobile

and immobile type dislocations:

αααα ρραρ gMIcM vR66 −= (22)

Combining all of the preceding considerations, the evolution rate of mobile dislocations is:

ρMα =α1ρM

α vgα

l− 2α2Rc ρM

α( )2vg −α3ρM

α vgl+α4

τ α

τ *

"

#$$

%

&''

r

ρIα vg

α

l+α5 pαβρM

β

β

N

∑vgα

l−α6RcρI

αρMα vg

α

(23)

The aforementioned mechanisms of mobile dislocations evolution, including Eq. (19),

(20), and (22), also involve the immobile dislocations. Based on these, the evolution rate of

immobile dislocation density is:

ρIα =α3ρM

α vgα

l−α4

τ α

τ *

"

#$$

%

&''

r

ρIvgα

l−α6RcρI

αρMα vg

α

(24)

The predicted mobile and immobile dislocation density are used to calculate the total

dislocation density using Eq. (16), which in turn predicts the resistance from dislocation

hardening according to Eq. (12).

The evolution rate of statistically stored dislocation density along slip system α is

treated as the summation of the two dislocation density evolution rate above, similar to Eq.

(16):

ααα ρρρ MIS += (25)


In our current framework, another kind of dislocation, geometrically necessary

dislocation, is introduced to taken into consideration of strain gradient for the long distance

interaction. The evolution rate of total dislocation, ρ , is composed of two parts, evolution rate

of GND, Gρ and the evolution rate of SSD, Sρ :

GSGS ρρρρρ α +=+= ∑ (26)

In traditional strain gradient theory, the magnitude of the plastic strain gradient is of the

order of the average shear strain in the crystal divided by the local length scale λ of the

deformation field. The strain gradient is γ/λ, where γ is macroscopic plastic shear strain. In

approximate terms,

λγ

ρbG4

= (27)

And the flow stress, or macroscpic shear yield stress is approximately

SGCGb ρρτ += (28)

where G is the shear modulus, b is the magnitude of the Burger’s vector, and C is the constant

taken to be 0.3 by Ashby (1970).

In our framework, this geometrically necessary dislocation is calculated out by:

( ) ( )2,

2,

ααα ρρρ screwGedgeGG += (29)

where )(

)(

, ~1

n

n

edgeG b ξγ

ρα

∂∂

−= and

)(

)(

, ~1

n

n

screwG b ζγ

ρα

∂∂

=

Here ξ and ζ denote, respectively, directions parallel and perpendicular to the slip direction on

the slip plane.


3. STRAIN GRADIENT FIELD CALCUALTED FROM POLYCRYSTALLINE VISCOPLASTICITY MODEL

To simulate the deformation behavior of polycrystalline materials, we used a

viscoplasticity self-‐consistent model (Lebensohn and Tomé, 1994). As mentioned previously,

fully plastic deformation is assumed, and it is further asserted that the deformation takes place

through shear and is independent of the hydrostatic stress. This allows a five dimensional

vector space to be used to in the formulation of inelastic deformation in terms of conjugate

deviatoric stress and strain rate tensors. Interchanging two components of the stress and

strain convention adopted by Lequeu et al. (Lequeu et al., 1987), the stress and strain rate

vectors from second order tensor are given by

{ }T

33 11 33 22 22 1123 13 12

( ) ( )' 2 , , , ,22 3

σ −σ + σ −σ σ −σ⎧ ⎫σ = σ σ σ⎨ ⎬

⎩ ⎭ (25)

{ }T

33 11 33 22 22 1123 13 12

( ) ( )2 , , , ,22 3

ε −ε + ε −ε ε −ε⎧ ⎫ε = ε ε ε⎨ ⎬

⎩ ⎭

& & & & & && & & & (26)

The scalar product of these two vectors gives the stress power, i. e.,

ijijkk εσεσ '' = (27)

where it is understood that the sum on subscript k is over the range 1,2,...,5. Inelastic

deformation occurs only when a slip and/or twin system is active. Both slip and twin systems

are characterized by two vectors: the slip systems unit normal vector n and the unit vector b in

the slip direction. Again, twinning is treated in this formulation as pseudo-‐slip, with the same

formulation of the kinematics as crystalline slip.

Each grain of the polycrystal has the kinematic relation provided by CDD. The VPSC

model couples the strain-‐rate and the stress in each grain with the average strain rate and

stress of the polycrystal (Lebensohn and Tomé, 1993). Each grain is regarded as an

inhomogeneity embedded in the homogeneous equivalent medium (HEM) having a viscoplastic

compliance (tg)M and a reference strain rate 0E& , whose behavior is identical to the average of

the polycrystal with a viscoplastic compliance c(tg)M and a reference strain rate 0&ε . This


inhomogeneity has a local stress field when a uniform stress is applied to the HEM. Using the

Eshelby formalism, the inhomogeneity is replaced by an ‘equivalent inclusion’ having the same

moduli as the polycrystal, subjected to a fictitious transformation strain rate *&ε .

The overall viscoplastic compliance moduli of the grain, (sec)M , can be calculated in a

self-‐consistent iterative way, as the HEM is assumed to describe the average behavior of the

aggregate. This forces the weighted average of stress and strain rate over the grains to coincide

with corresponding macroscopic values.

Texture evolution is simulated by enforcing the polycrystal deformation through

successive incremental steps. These are obtained by imposing a macroscopic strain rate E&

during the time interval tΔ , with a guess value at each step used for the strain rate &ε in each grain. The first deformation step uses a Full Constraints guess. The stress is then calculated

with each following step using the values from previous steps as the starting guesses. The

macroscopic secant modulus (sec)M is estimated using the Voigt average, (sec) 1 c(sec) 1− −=M M

for the first step, with the following steps using preceding values to derive the next estimate.

This modulus is then used to calculate the Eshelby tensor, S, the interaction tensor, M%, and the

accommodation tensor, cB . The average c(sec) cM B is used as an improved guess for (sec)M ,

with repeated iterations until the average coincides with the input tensor, under a certain

tolerance. This value is then used to calculate the macroscopic stress

(sec) 1' −=M E&Σ (28)

Each grain is then allowed to reorient due to slip and twinning, following a convergence

criterion. The lattice rotation rate for each grain is given by

1 s s s s s

ij ij ijkl klmn mn i j j is

1S (b n b n )2

−ω =Ω +Π ε − − γ∑%&& & & (29)

where Ω is the anti-‐symmetric component of the macroscopic distortion rate, Π is the

reorientation of the associated ellipsoid, and S is the antisymmetric component of the plastic

distortion rate (plastic spin); Π is proportional to the difference between the strain rate of the

grain and that of the polycrystal. Tiem et al. (Tiem, 1986) showed that for the elastic inclusion

case, Π increases with ellipsoid distortion. The modeling of the grain orientation will then use


the Volume Transfer Scheme as described by Lebensohn and Tome (Lebensohn, 1991; Tome,

1991) for twinned volume fractions. The polycrystal is represented as a discrete set of

orientations. These orientations are fixed, but their representative volume fractions evolve

during deformation. The Euler space is partitioned regularly in equiaxed cells or bins of 10

degrees on dimension of Euler angle. The orientations coincide with the centers of the cells. A

certain volume fraction of the material is assigned to each cell, corresponding to the initial

texture. The reorientation of lattice orientation that drives texture evolution during inelastic

deformation is visualized as displacements in Euler space of the represented points. If the cell

as a whole displaces rigidly and partially overlaps neighbors, then the volume fraction of

material in the overlap is subtracted and transferred to the neighbors. This is repeated every

iteration, providing gradual texture evolution.

4. RESULTS AND DISCUSSION

4.1 Low Length Scale Models Providing Dislocation Density Evolution Law

While this work focus on continuum dislocation dynamics, some example of molecular

dynamics and discrete dislocation dynamics simulation results is presented here to

demonstrate the flow of the information from microscale to mesocale.

In the dislocation dynamics framework, the mobility of a screw dislocation is considered to

be a small constant fraction of the mobility of the edge dislocation therefore, if the mobility of

either an edge or a screw dislocation is known, the mobility of the other type can be calculated

in terms of this mobility (Gilbert et al., 2011; Marian and Caro, 2006). Therefore, if the mobility

of the edge dislocation is known, the mobility of the screw can be also found. Results for the

dislocation mobility in iron alloys have been reported elsewhere in (Lim et al., 2011; Mastorakos

et al., 2010; Mastorakos et al., 2011a; Mastorakos et al., 2011b). The Embedded Atom Method

(EAM) (Daw and Baskes, 1983) and potentials developed in (Bonny et al., 2009; Mendelev et al.,

2003; Stukowski et al., 2009) for the pure iron, Fe-‐Cr and Fe-‐Ni respectively were used. In


particular, the dislocation mobility in a series of Fe-‐Ni-‐Cr systems at different temperatures has

been simulated using molecular dynamics.

Figure 3. Dislocation mobility as a function of the temperature in Fe-Ni system, with concentration of nickel or chromium was varied from 5% to 20%

Representative results for the edge dislocation mobility, in Fe-‐Ni alloy with nickel

composition at 0%, 10%, 15% and 20%, are shown in Figure 3. The results from these

simulations lead to power law relationship for the dislocation mobility of the form

00 TTmaM −+= where M is the mobility, T the absolute temperature, a0, and m are

numerical parameters. a0 varies between 3771 to 5135, and m between 321 to 367. The results

reveal that edge dislocation mobility is higher in the Fe-‐Ni systems compared to the α-‐Fe and

increases as the Ni concentration increases. On the other hand, the dislocation mobility inside

the Fe-‐Cr is comparable to that of pure iron although it still is about 5% higher. The higher edge

mobility of the alloys compared to the pure Fe is backed by experimental observations that

show a higher density of screw dislocations in alloys because the edge dislocations are very fast

and disappear at the surface of the specimen (Guyot and Dorn, 1967; Nadgorny, 1988). This is

also backed by other simulations of screw dislocations mobility showing that it is lower than the

edge mobility (Gilbert et al., 2011; Marian and Caro, 2006)


a) b)

Figure 4. a) Dislocation-defect structure after plastic deformation. b) Stress-strain curves simulated by

DDD for iron single crystal at different defect densities (a:0, b:2×1022, c:5×1022, d: 7×1022, e: 1×1023 , f:

2×1023.)

Dislocation mobilities of iron alloys calculated from molecular dynamics are used in

conjunction with the dislocation dynamics simulations to predict the evolution of dislocation

density as a function of alloy and defect density. The DD simulation unit cell, shown in Figure

4(a), is a 4.5×4.5×5.97 µm3 cube cell that contains an initial density of Frank-‐Read sources

distributed randomly on the primary (Hiratani et al.) slip planes. In the simulations, periodic

boundary conditions were imposed. The cell is loaded in tension with a constant strain rate of

100/s. The effect of irradiation is accounted for by mapping into the DD box a spatial

distribution of FS loops with density ranging from 1020 /m3 to 1024 /m3. The loops are 1 to 3

nanometers in radius, and the radius is randomly generated to fall within the specified interval.

The model also generates Frank-‐Read sources, represented as finite dislocation segments

pinned at ends, lying on (Hiratani et al.) glide planes, and Burgers vectors of the type <111>.

Dislocation density evolution and the mobile-‐dislocation-‐related CRSS are predicted from

discrete DD. The dislocation structure shown in Figure 4a is composed mainly of extended

dislocations of screw character, resulting from the fact that the mobility of the screw

dislocation is three orders of magnitude less than the edge. As the dislocations sweep through


the cloud of the FS loops, some of the loops get annihilated, depending on their interaction

energy with the sweeping dislocations as discussed before by various authors (Ghoniem et al.,

2000; Hiratani et al., 2004; Khraishi et al., 2002a).

This results in the formation of defect-‐free channels, and subsequently causes a drop in the

flow stress as deduced from the stress-‐strain curves. Figure 4(b) demonstrates the initial

deformation behavior of pure iron with different defect densities. In all cases, the stress

initially increases linearly until it is high enough for dislocations to overcome internal barriers

and the pinning effect of the defects, which arises only from dislocation-‐FS loops elastic

interactions. As the dislocations propagate and interact with the defects, their interaction

energy with the defect at a critical distance can cause defect absorption within the dislocation

core, which, in turn, leads to a drop in stress (Khraishi et al., 2002b) and a decrease in defect

density.

(a)


(b)

(c)

Figure 5. Discrete dislocations dynamics results: Evolution of defect density (a) and dislocation density

(b) for iron alloy with 5%Ni concentrations at different initial irradiation defect density: (a:0, b:2×1022,

c:5×1022, d: 7×1022, e: 1×1023 , f: 2×1023.) , (c) evolution of total dislocation density, cross-slip segments

and dislocations junctions.


Figure 5(a) shows typical DD results for the evolution of the defect density with

deformation. In all cases, the defect density remains unchanged during the elastic loading as

expected. As the stress becomes high enough to cause the dislocations to move, the

dislocations interact with the defects and some of them get annihilated, causing a decrease in

the defect density. In the meantime, the dislocation densities in Fe-‐Ni alloys with different

initial defect density generally increase with deformation, as shown in Figure 5(b). Dislocations

are further categorized and recorded during the deformation process. Figure 5(c) provides the

detailed information of evolution of cross slip and junction. This information will be used in

calculating the dislocation density evolution parameters α1-6.

4.2 CDD in Predicting Mechanical Properties of Fe Single Crystal With the predicted dislocation density evolution from DDD, CDD is used in predicting

the flow behaviors of Fe single crystals. The advantage of this framework is the physical

meaning of all the parameters in the mesoscale model. With consideration of cross-‐slip,

anisotropic Peierls stress for different slip systems, CDD can predict the strength and

deformation behavior of single crystals with higher fidelity. The parameters for pure α-‐Fe used

are listed in Table 1, partially built upon Lee’s previous works (Lee et al., 2010b; Lim et al.,

2011).

Table 1 . List of parameters used in the continuum dislocation dynamics model Symbol Denotation Value Μ Shear modulus 80GPa Ν Poisson’s ratio 0.3 C11 Anisotropic elasticity constant 242 GPa C12 Anisotropic elasticity constant 150 GPa C44 Anisotropic elasticity constant 112 GPa γ0 Reference strain rate 4×10-‐5 m/s M Strain rate sensitivity 0.012 Α Baily-‐Hirsch hardening coefficient 0.4 B Magnitude of Burger vector 2.54×10-‐10m τ0 Peierls stress (internal friction) 11.0MPa Β Irradiation hardening coefficient 0 α1 Mobile dislocation multiplication coefficient 0.02


α2 Mobile-‐mobile dislocation annihilation coefficient 1.0 Rc Critical radius for annihilation in units of Burger vector 15 α3 Immobilization parameter, mobile to immobile 0.002 α4 Mobilization parameter, mobile to immobile 0.002 α5 Cross-‐slip coefficient 0.018 α6 Mobile-‐immobile annihilation coefficient 1.0

One point should be emphasized here: the parameters are either experimentally

measured (such as μ and C11) or calculated from the discrete DD (e.g., the parameters in the

mechanisms of dislocation density evolution law, α1, α2, etc.). Along with the advantages

already mentioned, there are four other salient points in the proposed framework. The first is

the application of cross-‐slip, the second is the anisotropic Peierls stress, the third is the updated

strain rate sensitivity law with information of dislocation density, and the fourth is the

capability to predict irradiation hardening by introducing hardening due to interaction between

dislocation and defects.

We compared the CDD prediction results with the experimental results measured by

Keh et al. (Keh, 1965) and simulation results from SCCE-‐T and SCCE-‐D. The CDD parameters

used in Table 2 are from predicted results using DDD and lower-‐length scale models. The

parameters used in SCCE-‐T and SCCE-‐D are from back-‐fitting the mechanical testing results. 12

(1 1 0)[1 1 1] and 12 (1 1 2)[1 1 1] slip systems in body center cubic (BCC) crystal systems were

considered for crystallographic slip. In the initial state, mobile and immobile defect density are

considered the same, and are distributed uniformly along 24 slip systems.

Using uniaxial tension stress strain curve of Fe single crystals along [100] direction as a

benchmark in SCCE-‐T and SCCE-‐D, the simulated flow curves along [100], [011] and [-‐348] are

shown in Figure 6. The process of back-‐fitting constitutive parameters in SCCE-‐T and SCCE-‐D are

detailed in previous works (Lee et al., 2010a). Using the parameters fit from [100] direction,

SCCE-‐D and SCCE-‐T both agree with experimental data along [100], even better than the

prediction results from CDD. However, using the same parameters to simulate the flow curve

along [011] direction, SCCE-‐D underestimates and SCCE-‐T overestimates the stress. When

applied on [-‐348] direction, SCCE-‐D overestimates more than 50% of the experimental results


while SCCE-‐D overestimated more than 100% of the experimental data. Predicted results using

CDD agree well with experimental data along all three directions.

(a) (b)

(c)

Figure 6. Comparison of experimental stress strain curves and simulated results from SCCE-T, SCCE-D

and CDD models for iron single crystal with uniaxial tensile loading direction along (a) [100] (b) [011]

and (c) [-348] directions. The constitutive law parameters for SCCE-T and SCCE-D are back-fit to the

[100] results.

On the other hand, if the constitutive parameters are back-‐fitted using the

experimental data along [-‐348] direction, SCCE-‐T and SCCE-‐D only work in the benchmark case,

[-‐348] shown here in Fig. 7(c). But they will not work in the other directions, as shown in Fig.


7(a) and (b). The large deviation demonstrates the limitations of empirical constitutive

equations without scientific mechanisms in background support.

(a) (b)

(c)

Figure 7. Comparison of experimental stress strain curves and simulated results from SCCE-T, SCCE-D and CDD models for iron single crystal with uniaxial tensile loading direction along (a) [100] (b) [011] and (c) [-348] directions. The constitutive law parameters for SCCE-T and SCCE-D are back fit to the [-348] results.

Experimental data of single crystal iron demonstrated large behavior differences when

uniaxially stretched along different directions. The yield stresses are similar for all directions,

around 35 to 40 MPa. Contrasting to the large strain hardening with the stress increased

dramatically to over 90MPa along [100] direction, there is little strain hardening when

stretched along [-‐348] direction. This is due to the number of slip systems activated. Fig. 8


shows the evolution of mobile dislocation density (Fig. 8a) and total dislocation density (Fig.

8b). With the increase of strain, the dislocation density along (-‐211)[111] increases with small

vibration due to annihilation and transfer to immobile type. Dislocation densities along all the

other directions keep almost constant to the end. Combined with the information from

immobile dislocation, the total dislocation density increased smoothly with strain, compared

with the evolution of mobile dislocation density along (-‐211)[111].

Some parameters used in CDD may be measured directly from experiments; some from

DDD and MD simulations. Parameters like cross slip coefficient is a parameter hard to measure

from experiment, while difficult to simulate directly since it is sensitive to the dislocation

microstructure. It is the same to Peierls stress anisotropic factor. A parameter sensitivity study

has been carried out to conduct the comparison of the variance with different CDD parameters.

Experimental stress strain curve of iron single crystal is used to verify the choice of parameters.

The table below shows a part of the parameter fit results we obtained for the variance:

Table 2. Variance of the simulation results from experimental stress strain curve for two CDD parameters

parameters

Peierls stress anisotropic factor=1 1.5 2 2.5 3.5 4 4.5

cross slip coefficient=0 0.514 0.6211 0.6153 0.6056 0.6056 0.6056 0.6056

0.004 0.5094 0.5353 0.4708 0.4641 0.4637 0.4626 0.4655 0.008 0.4407 0.3864 0.3239 0.3189 0.3177 0.3229 0.3199 0.012 0.2976 0.2138 0.1749 0.1861 0.1739 0.1725 0.1779 0.016 0.1284 0.0562 0.064 0.0523 0.0434 0.0472 0.049 0.02 0.0525 0.0334 0.0433 0.0448 0.0436 0.0397 0.0343

For a robust simulation, cross lip coefficient of 0.02 and Peierls stress anisotropic factor

of 1 are the best from the simulated results demonstrated in Table 2.


4.3 Prediction of Deformation Behavior and Texture Evolution using CDD-CP

Constitutive law based on dislocation density evolution predicted from continuum

dislocation dynamics is introduced into the viscoplasticity model to solve the boundary

condition loading stress and strain. This is in turn passed into the continuum dislocation

dynamics model to solve the evolution of mobile and immobile dislocation density in each

crystal and the corresponding slip resistance for each slip system.

CDD-‐CP is applied in a polycrystalline iron alloy with bcc crystal structure and random

texture, composed of 100 crystals. (100), (110) and (111) pole figures of the selected system is

demonstrated in Fig. 8. The maximum texture intensity in them is lower than 1.3 times random.

Figure 8. (a) 100, (b) 110, and (c) 111 pole figures of simulated bcc iron sample with random texture.

Under uniaxial tension, the predicted stress strain curve up to a strain of 10% is presented

below, with features of strain hardening and saturation hardening captured.

The stress flow in Fig. 9 demonstrated yielding around a strain of 0.02. After that, strain

hardening started to saturate. Texture evolution during uniaxial tension is shown in Fig. 10:


Figure 9. Predicted stress strain curve of the random texture iron using CDD-CP model.

(a)

(b)

Figure 10. Predicted (100), (110) and (111) pole figures of random bcc iron alloys under uniaxial tension

at strain of (a) 2% and (b) 10%.

0

10

20

30

40

50

60

70

0 0.02 0.04 0.06 0.08 0.1

true

stre

ss (M

Pa)

true plastic strain

stress strain curve of polycrystalline iron alloy predicted

TDMD


In the (100) pole figure, a texture component with (001) aligned along machine direction

(tensile here) appeared at a strain of 2% and strengthened at 10%.

A simulated microstructure with single crystal grid was generated with random texture.

Each grid represented a single crystal with different orientation. The heterogeneous

microstructure introduce different strain field, with a strain gradient field. Below is a work flow

for the closely coupled crystal plasticity model and CDD. In the gradient plasticity CDD informed

CP, the feed from CP has strain gradient added while the return from CDD to CP has more

information on the GND, SSD and updated constitutive law.

(a)

(b)

Figure 11. Scheme of data flow in closely coupled CDD and CP (a) without (b) with long distance interaction considered.

The simulation results for the strain of a random microstructure are illustrated in the figure

below. There are 10×10 single crystals, the strain of each grain, or grid, is represented by a color

grain when the sample is uniaxial tensioned to a different effective strain in Fig. 12.

CDD CP

Constitutive law, dislocation density

Stress, strain, boundary condition


(a) (b) (c)

(d) (e) (f)

Figure 12. Strain field of random textured polycrystalline agglomerate after uniaxial tensioned to a strain of (a) 0.1, (b) 0.2 (c) 0.3 (d) 0.5 (e) 0.8 and (f) 1.0.

Correspondingly, the strain gradient calculated from the above strain field is illustrated in the

figure below.

(a) (b) (c)


(d) (e) (f)

Figure 13. Strain gradient field of random textured polycrystalline agglomerate after uniaxial tensioned to a strain of (a) 0.1, (b) 0.2 (c) 0.3 (d) 0.5 (e) 0.8 and (f) 1.0

When a large system with 2000 crystals, a grid of 40×50, is used, the distribution of strain is

more uniform, as shown in figure below.

(a) (b) (c)

(d) (e) (f)

Figure 14. Strain field of random textured polycrystalline agglomerate with 2000 grains after uniaxial tensioned to a strain of (a) 0.1, (b) 0.2 (c) 0.3 (d) 0.5 (e) 0.8 and (f) 1.0

The responding strain field calculated is illustrated below:


(a) (b) (c)

(d) (e) (f)

Figure 15. Strain gradient field of random textured polycrystalline agglomerate with 2000 grains after uniaxial tensioned to a strain of (a) 0.1, (b)0.2 (c) 0.3 (d) 0.5 (e) 0.8 and (f) 1.0

Further comparison with experimental data is necessary for validation.

5. IMPLEMENTATION OF CRYSTAL PLASTICITY IN MARMOT

A simplified version of the CDD model was also implemented as a material calls

within MARMOT, a multi-‐physics, finite element simulation tool focused on the mesoscale.

MARMOT is implemented using the MOOSE framework from INL (Tonks, 2012).

As discussed in Section 2, our constitutive law for the shear strain rate is based on the

Orowan relation (Orowan, 1940):

ααα ργ gMbv= (30)


where αρM is the mobile dislocation density of slip system α, b is the Burgers vector, and α

gv is

the average dislocation glide velocity in slip system α.

As in previously developed models, for our simplified CDD model the dislocation glide

velocity αgv , is expressed in a power law of resolved shear stress ατ similar with shear strain

rate of slip system α, ατ 0 :

( )αα

αα τ

ττ signvv

m

g

1

00= (31)

It should be noted, this is a deviation from our CDD model implementation in the VPSC

framework. The dislocation glide velocity expression shown in Eq. (10) will be implemented in

future work. As shown in Eq. (11), the slip resistance of slip system α, ατ 0 , is a summation of

reference resistance τ0 from lattice friction stress for moving dislocations; resistance due to

dislocation-‐defect interaction τdα, mainly from irradiation; and resistance from dislocation

hardening ατ dh :

ααα ττττ dhd ++= 00 (32)

The dislocation hardening term, ατ dh , is the resistance of statistically stored dislocations on

each slip system from moving dislocation on the system n.

m

m

mdh b ρµατ αα ∑ Ω=

(33)

where mαΩ is the interaction matrix of the slip systems and ρm is the summation of the

mobile and immobile dislocation densities. For this initial simplified implementation, the

dislocation density term excludes the strain gradient influence.

For the VPSC implementation, defects are assumed to be distributed randomly for all slip

systems and a modified dispersed barrier hardening model (Eq. 14) is used to express

irradiation resistance ατ d from dislocation-‐defect interaction. For this simplified


implementation, the defect density and defect size are held constant and available as model

inputs. Therefore ατ d , becomes constant for a given simulation.

The statistically stored dislocations required by the dislocation hardening term can be

generally divided into two types, mobile and immobile, as shown in Eq. (16). The evolution

equations for both types of dislocations are shown in Eq. (23) and (24), respectively. See

Section 2 for further discussion of the evolution equations.

5.1 MARMOT Implementation

The CDD implementation in MARMOT begins from the strong form of the governing

equations on the domain Ω and boundary Γ = ΓliΓgi as follows:

∇⋅σ + b = 0 in Ω (34)

u = g in Γg (35)

σ ⋅n = l in Γl (36)

where σ is the Cauchy stress tensor, u is the displacement vector, b is the body force, n is

the unit normal to the boundary, g is the prescribed displacement on the boundary, and l is

the prescribed traction on the boundary. The weak form of the residual becomes:

ℜ = σ ,∇φm( )− l,φm − b,φm( ) = 0 (37)

where ⋅( ) and ⋅ represent volume and boundary integrals, respectively. The Jacobian

required by Newton’s method for solving the residual equation can be expressed as the

following, when ignoring the boundary terms:

ℑ =

∂σ∂∇u

,∇φm$

%&

'

() (38)


The stress is a nonlinear function of the strain during plastic deformation. This function is

defined by crystal plasticity theory (see section 2, Eq. 1-‐5) and our simplified CDD constitutive

model as described in the previous section.

There are a series of material base classes available through MARMOT. The CDD model was

implemented using the FiniteStrainMaterial base class that is part of TensorMechanicsMaterial

implemented in ELK, another component of MOOSE. All material base classes provide methods

for initializing state properties, calculating incremental strain updates, and calculating the

current stress. The CDD implementation utilizes the existing strain update method while

overriding the initialization and stress calculation methods.

The method computeStrain follows the algorithm outlined by Rashid (Rashid, 1993) for

calculating the incremental deformation gradient. The method computeQpStress is left pure

virtual for any material that inherits to override with the desired constitutive relationship.

Finally, the method initQpStatefulProperties can be overridden and expanded to initialize

additional state variables as needed.

The computeQpStress implementation used here follows the patterns seen elsewhere in

MARMOT.

ε e = εne +Δε

Dp = εnp

solveStressResidual(σ n,Δε,Cijkl,Dn+1p ,σ n+1)

εn+1p = RDn+1

p RT

σ n+1 = Rσ n+1RT

(39)

where the elastic strain, 𝜀!, and plastic strain, 𝜀!, are first updated. Then the stress residual is

solved. This method updates values of the plastic strain rate, 𝐷!!!! , and stress, 𝜎!!!. Before

exiting, rotations are applied. The bulk of the constitutive model is implemented within the

method solveStressResidual.

solveStressResidual integrates the CDD constitutive model in an incremental manner using a

Newton Raphson implementation. Given the complexity of the constitutive model this is


implemented in two levels. An internal loop solves the stress residual while an external loop

updates and solves the slip resistance residual.

5.2 Plan for Strain Gradient Addition to MARMOT Implementation

The incorporation of a strain gradient term to the CDD model will enable this crystal

plasticity model to more accurately capture the effects of long-‐range dislocation interactions.

While the evolution equations for immobile dislocations presented above account for short-‐

range interactions among dislocations, these evolution equations do not at present include the

effects of long-‐range interactions. Long-‐range forces also act to impede dislocation motion. In

the vein of the work of Fleck and Hutchinson (Fleck and Hutchinson, 1997; Fleck et al., 1994;

Fleck and Willis, 2009a), and following the work of Zbib and Aifantis (Zbib and Aifantis, 1992,

2003), we introduce the use of a strain gradient term to capture long-‐range geometrically

necessary dislocations (GNDs).

Long-‐range forces resulting from GNDs are an important source of hardening. In the

formulation used here, the gradient of the plastic strain with respect to both the normal and

tangential in-‐plane directions is calculated, representing edge and screw GNDs, respectively.

The plastic strain values are equivalent to the slip in each crystallographic slip system.

Multiplied by a length scale term, these strain gradient values are used to update the mean free

dislocation glide path term:

ρIα =

Δγα

bLα, where Lα = c*

ωαβ ρIβ + ρGND

β( )β

∑ (40)

where, 𝐿! is the updated mean free path term. The mean free path term is used in the

evolution of immobile dislocation equations. Inclusion of the strain gradient term allows for a

more accurate hardening model.

Numerically the strain gradient term is acquired as the derivative of strain values collected

at the integration points of several elements. It is critically important that the strain gradient

term depend on strain values from more than a single element. Methods that calculate the


strain gradient within a single element through a second derivative of the shape functions, such

as work by Busso et.al. (Busso et al., 2000), fail to capture the long-‐range nature of GND

interactions. While these methods are computationally compact, these formulations limit the

range of GND interactions. The potential for the introduction of numerical artifacts also exists

within these single element methods. Furthermore these methods, which calculate the strain

gradient within a single element, produce a mesh-‐dependent strain gradient value.

Employed here is a formulation that links the strain gradient term to strain values across a

cumulative volume of elements. The use of multiple elements in the calculation of the strain

gradient term enables a mesh-‐independent strain gradient: the long-‐range effects captured in

this manner are not bound by the size of elements.

Concerns about the continuity of the strain are an issue with which the strain gradient

calculation must deal. Previous work in the field has made use of meshless methods or higher

order (C1) formulations to reach across element boundaries. As discussed below in greater

detail, care is taken to address continuity when using strain values from multiple elements.

Original efforts to implement a strain gradient term into the CDD model focused on

leveraging previous work within the context of an ABAQUS UMAT at Washington State

University (WSU). Unfortunately direct porting of the strain gradient calculation from the

UMAT was not possible: the absence of key utility functions in MOOSE and the heavy

computational effort required by the WSU UMAT method forced the exploration of C1

formulations. Thermodynamic consistency and straightforward integration with existing CDD

formulations were of primary consideration in selecting and deriving an appropriate C1

formulation. For developmental ease, the isotropic hardening small strain C1 formulation is

presented below.

The previous implementation utilized a moving weighted least squares regression method.

Through the use of a computationally expensive ABAQUS utility function, strain values from

across multiple elements are used to calculate the strain gradient. Integration points from

multiple elements are selected within a radius of capture, and the strain values from these

selected integration points are numerically collected with the utility function. These collected


strain values are fitted to a second order polynomial curve using a Moving Weighted Least

Squares Regression (MWLSR); the moving description refers to the use of the radius of capture

to select integration points. For each integration point in the model a new selection process is

performed by sweeping a sphere, defined by the radius of capture, around the integration

point.

Following the work of Abu Al-‐Rub et al. (Abu Al-‐Rub et al., 2007), a weighting function is

used to assign more importance to the fit of strain values from integration points near the

center of the sphere. As a result of this method strain values within the same central element

are weighted more heavily by the MWLSR method than strain values from outer elements. The

reasoning of the use of the weighting function is threefold: 1) the weighting function provides a

more stable numerical solution, 2) the weighting function in some way mimics the 1 𝑟 behavior

of long-‐range dislocation interactions, and 3) the weighting function provides a limited ability to

address continuity issues. As noted by de Borst et al. (De Borst and Pamin, 1996), only the

discretization of the gradient term requires C1-‐continuous shape functions. In the vein of

meshless methods, the weighting function is an attempt to provide continuity in the strain

gradient calculation despite the use of strain values from several C0 elements.

The derivative of the second-‐order MWLSR produced curve, with respect to the normal and

tangent directions, is taken as the strain gradient for each of the slip systems in the crystal.

Using this method two strain gradient terms are calculated for each slip system, increasing the

computational effort required.

This procedure, used to calculate the strain gradient in ABAQUS, is extremely computational

expensive. The obvious main source of computational load is the MWLSR, which is completed

twelve times for every integration point in the mesh of an FCC crystal system. Contributing

factors also include the calculation of selected integration points via the radius of capture and

the use of the utility function to call the strain values at all of the selected integration points.

The error introduced by the MWLSR process should also not be disregarded; this error is carried

forward into the crystal hardening.


Although the strain gradient term would be the most useful in polycrystalline simulations,

the computational effort required by this implementation severely limits the number of crystals

that can be included in an ABAQUS simulation.

In light of these complications with the strain gradient calculation method used in ABAQUS,

we elected to explore the option of a higher order C1 formulation. The decision to move to the

C1 formulation was affirmed by the absence of a MOOSE utility function to numerically collect

strain values from several integration points. The higher order C1 formulation proposed for

implementation into the CDD model in Marmot is more strongly rooted in thermodynamic

principles. Following the strain gradient reformulation done by Abu Al-‐Rub, this C1 formulation

is admissible under the Clausius-‐Duhem Inequality when non-‐local quantities are properly

considered.

Since the C1 formulation is continuous in both displacement and strain, this formulation has

the potential to be more accurate than MWLSR method currently employed in the ABAQUS

UMAT at WSU. Since the strain is continuous across neighboring elements, it is not necessary

to use a weighting function to force continuity for the strain gradient calculation. The C1

method could also eliminate the error resulting from the curve-‐fitting, if the additional degrees

of freedom (moments) are adequately treated.

In the literature comparisons (Huang et al., 2004) of C0 and higher order formulations have

noted that the primary difference between these two categories of formulations is seen at

interfaces. Based on perturbation work performed by (Shu et al., 2001), the effect of higher

order formulations is evident only in a thin layer near boundaries. While many groups have

often neglected this thin boundary layer, capture of this boundary layer could prove to be

useful in future multi-‐scale modeling efforts. We would argue that use of the more complex C1

formulation is a responsible choice for enabling future model development.

Finally, and critically, the C1 formulation would not require the use of a computationally

intensive utility function to call strain values from multiple elements. Rather the C1 formulation

would make use of the additional degrees of freedom available in the third-‐order Hermite

element to calculate the strain gradient.


The challenges associated with a C1 formulation should not be overlooked. As discussed by

Zhang et al. (Zhang et al., 2013) in a phase field application, a C1 formulation significantly

increases the stiffness matrix bandwidth. This increased bandwidth results in a longer

simulation wall clock time. At present, only a limited number of hermite elements are available

in the libMesh library; therefore, only simple geometries can be studied with a higher order

formulation for now. Proper treatment of the higher order terms introduced in a C1-‐continuous

yield condition is not trivial. Any development work considering a C1 formulation should

carefully weight the potential increased accuracy against the certain increase in computational

expense.

5.2.1 Theoretical Overview of Higher Order Strain Gradient Formulation

A brief overview of the theory behind the proposed C1 formulation to calculate the strain

gradient is presented here. Several excellent reviews of the current field of strain gradient

work are available in the literature (Abu Al-‐Rub et al., 2009; Evers et al., 2004; Fleck and

Hutchinson, 1997; Huang et al., 2004; Roters et al., 2010); however, we only discuss key

concepts for brevity's sake.

We motivate the use of the strain gradient to account for GNDs through the definition of

the Nye's tensor (Arsenlis and Parks, 1999; Bassani, 2001; Nye, 1953). The Nye's dislocation

density tensor is a general representation of the i-‐component of the net Burger's vector related

to the GNDs with j direction:

αij = ρGNDβ

β∑ bi

βt jβ (41)

, where 𝜌!"#! is the GND density, 𝑏!

! is the Burger's vector, and 𝑡!! is the tangent unit vector for

the ## slip system. The use of the generalized Nye's tensor allows for the accurate non-‐uniform

distribution of GNDs through the crystal geometry. Following (Arsenlis and Parks, 1999;

Bassani, 2001; Fleck and Hutchinson, 1997), the definition for the Nye's dislocation density


tensor can be written as a function of the second derivative of displacement through

application of the Stokes' theorem.

αij = ejklui,lkp (42)

Because of the choice to use the Hermite third-‐order element, the second derivative of

displacement ui,lkp could be accessed directly via the second derivative variable class in MOOSE.

However, to ensure thermodynamic consistency, the second derivative of displacement can be

related to an effective plastic strain variable through the plastic strain

p = 23εijp εij

p =23uijp uij

p

(43)

Substituting the definition of the effective plastic strain from Eq. 43 into Eq 42, we derive

the Nye's dislocation density tensor as a function of effective plastic strain

αij = pkβ

β∑ ejklµil

β (44)

as in Gao et.al. (Gao et al., 1999). The introduction of the effective strain gradient to

incorporate additional isotropic hardening has been employed by several groups in various

forms (Acharya and Bassani, 2000; Muhlhaus and Aifantis, 1991; Polizzotto et al., 1998). Gurtin

(Gurtin, 2000, 2003) treated dependence on plastic strain gradients through microforce

balances and supplemented classical boundary conditions with non-‐local boundary conditions

from the flow rule.

Within the CDD model represented by equation Eq. 40, the scalar accumulation of the GND

density is used. By taking the square root of the square of equations Eq. 41 and Eq. 44, we

propose that the accumulated density of GNDs can be calculated from the effective plastic

strain by setting these two equations equal to each other:

α = αijαij = bρGND = ρ,kβρ,k

β

β

∑ , (45)


when interactions among dislocations on different slip systems are neglected. A similar

approach to an effective strain gradient is employed by Gao (Gao et al., 1999). Eq. 45

demonstrates the calculation of the density of the GNDS from the gradient of the effective

strain. This calculated effective GND density can then be used to update the mean free

dislocation glide path (Eq. 40 within the larger CDD model. The use of an accumulated GND

density value is in contrast to the tensor approach used by Akasheh et.al. (Akasheh, 2007 #100)

although this difference in approach should be evident only in the numerical implementation.

The effective strain gradient model we use here, differs from previous approaches (Acharya

and Bassani, 2000; Muhlhaus and Aifantis, 1991; Polizzotto et al., 1998) in the inclusion of the

effective strain gradient in the internal power expression. The principle of virtual power is used

to derive the microforce balance in the vein of Gurtin (Gurtin, 2003).

Considering higher order variables, introduced as the work conjugate to the gradient of

effective strain as done by Shu et al. (Shu et al., 2001), the virtual power balance is written for a

body V.

σ ij, jV∫ δ uidV + Ti −σ ijn j( )

∂V∫ δ uidS + τ ijNij − R+Qk,k( )

V∫ δ pdV + m−Qknk( )

∂V∫ δ pdS = 0

(46)

In this expression N is the flow direction, R represents the nonlocal dislocation drag stress

related to isotropic hardening, and Q is the higher order nonlocal work conjugate of the

effective strain gradient. This virtual power balance expression is based on the concept that

power expended can be associated with a force system; therefore, independent treatment of

the coupling of the nonlocal variables may not be needed.

Following the usual treatment, the boundary conditions can be determined from the virtual

power balance expression by requiring that variational displacement and effective strain (u and

p) may be arbitrarily specified. On the macroscopic level the classical boundary conditions

remain unchanged:

σ ij,i = 0 and Ti =σ ijn j (47)


Additionally two nonlocal boundary conditions also appear: the nonlocal microforce

condition

τ ijNij = R−Qk,k (48)

and the microtraction bound condition

m =Qknk (49)

The nonlocal microforce boundary condition may be considered as the nonlocal yield

condition, following Gurtin (Gurtin, 2003), and a nonlocal application of the Clausius-‐Duhmen

inequality may be used demonstrate thermodynamic admissibility of the associated

constitutive equations (Abu Al-‐Rub et al., 2007). The microtraction boundary condition is

considered to represent the interaction of long-‐range dislocation forces across interfaces, such

as the grain boundaries.

The use of a C1 formulation to incorporate the effects of long-‐range dislocations into the

CDD model is proposed as a solution to the numerical challenges and mathematical limitations

of the C0 formulation currently employed within the WSU ABAQUS UMAT subroutine. Although

this proposed C1 formulation is not without its own numerical implementation concerns, the

potential to capture the effects of long range GND densities is exciting

ACKNOWLEDGEMENTS

This work was funded by DOE’s Nuclear Energy Advanced Modeling and Simulation

(NEAMS) program at Pacific Northwest National Laboratory (PNNL). PNNL is operated by

Battelle for the DOE under contract No. DE-‐AC05-‐76RL01830.


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